where \(S_b(x)\)  is the double sine function [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, \(u_i\) , \(v_i\) , \(\xi _i\) , \(\zeta _i\)  are linear functions of \(a_i\) , \(b\) , \(a\)  and \(a^{\prime }\) , and the choice of integration contours is quite tricky [7]}, [8]}.
